From a modeling viewpoint, it is desirable that such important structural properties as energy conservation
are inherited by the approximation. Aside from interest in the Dysthe equation as an asymptotic model for water wave applications,
this question is particularly relevant considering the successful and widespread use of the Hamiltonian formalism
in physical sciences, including the field of fluid mechanics and free-surface flows [1]}, [2]}.
The associated mathematical tools as applied to this physical problem are thus also of interest.
Throughout our derivation, care is taken to perform both the expansion of the Hamiltonian functional
and the canonical change of symplectic structure in a systematic manner,
starting from the basic formulation introduced by [3]}.
Our calculations are facilitated by the fact that the Dirichlet–Neumann operator admits an explicit Taylor series expansion [4]}.
This property has been extensively used in previous studies on irrotational water waves [5]} and can also be exploited here.
